The short version of this article is that the two most common pictures of how the Higgs vacuum causes mass are very misleading. These are the drag or mud picture, and the particle being deflected by a series of Higgs virtual interactions picture. Why not just say that the Higgs coupling with an elementary particle is just the same as the mass term in the energy, and the constant Higgs vacuum value v, times its coupling strength to that particle can be used as its mass.
The drag theory of multiple virtual Higgs collisions or “mud” of course violates Newton’s law that a particle in motion remains in the same motion if not acted upon by an outside force. It actually leads into Aristotle’s fallacy based on observation that anything in motion comes to rest unless propelled by a force to overcome friction.
The second questionable explanation is a picture of a particle with a well defined momentum colliding with an X (or virtual Higgs interaction), and then following an new well defined momentum direction, and this continuing. Several things are seriously wrong with this picture. The first is just common sense. If a serious of random collisions hits a particle of definite momentum or direction of travel, the course it is following deviates by the square root of the number of collisions times the average change in momentum of each collision. This of course violates conservation of momentum and also energy, by a constant vacuum no less.
In that picture, which I’ll call the X picture, the first thing you learn in calculating Feynman diagrams (which that is supposed to be), is that energy and momentum are exactly conserved at each interaction, no matter how short in time or small in space they occur. Otherwise, the overall energy and momentum of the process could never come out the same as it started. The other problem, is the X does not have an emitted even virtual particle to conserve the energy momentum.
An adjoined problem of both pictures is that they are often accompanied by a serious misstatement of the uncertainty principles. Namely, that for a short time you can violate energy conservation, or for a short distance you can violate momentum conservation. Again, once violated by these interpretations, how do you get back to the initial energy and momentum for its conservation.
A truer path to acceptance is the real uncertainty principle. To specify the momentum of a particle exactly, its probability of observation must be spread uniformly over all space. To specify the energy exactly, the probability must be uniform in time. The Higgs vacuum value v is a constant over all space and time. This is consistent with the mass of a free particle being constant in space and time, and the desire for a microscopic mechanical model should perhaps be phased out.
Another description of the Higgs Field appears in a glossary by the New York Times on March 5, 2013. I applaud the NY Times for covering the Higgs and the experimenters and the state of US particle physics so extensively, and these comments are not meant to demean that excellent newspaper in any way. In the glossary, they describe the Higgs Field as a “field of energy”. This seems tricky to me to work with. It is true that the vacuum expectation v of the Higgs field φ (phi) has the dimensions of energy. But the potential energy of the Higgs field when it attains that minimum value of energy is zero, as shown in the picture of its potential below:
A field is really like a quantum mechanical wave function, which means that it is an amplitude who’s square gives the probability of finding the particle of the field there. It is only when you multiply that probability by the square of its mass that you get the energy density associated with that boson. The Higgs field can carry an energy, associated with the phase of the amplitude changing in time, but its vacuum expectation value v is taken as real, and it is therefore energy-less. This again appears to be a case of trying to describe the Higgs field as something else that we might understand, but is actually misleading.
What we can say about the Higgs field is that it has no electric charge, no intrinsic angular momentum or spin, no color force charge, and no odd parity. The vacuum expectation value of the Higgs field is constant in space and in time, so it has no momentum or energy. Is it the nowhere particle? Partly no, since it comes from a doublet and anti-doublet in weak isospin, so it has weak isospin I = 1/2, and Iz = -1/2 for the H0, and Iz = +1/2 for the anti-Ho. See my article on this under the category Higgs Boson on this blog.
The NY Times, in a great set of articles on the discovery of the Higgs vacuum field, called it an “energy field”. As we have said, the non-zero vacuum value of the field is only described by its real number value v. All particles and fields can carry energy by having a phase that oscillates in time with a rate given by their energy, that is they have a factor:
Cos(E t /hbar) + i Sin(E t / hbar),
where i is the imaginary unit number, and hbar is Planck’s constant h divided by 2π. The Higgs field vacuum value v is real and unchanging in time, and thus corresponds to zero energy. In fact, it is really the only zero energy field part around, not an “energy field”.
The other description in the NY Times is more appropriate, that the Higgs field is a “force field”. Bosons that are exchanged such as photons and gluons are often called force fields, and the particle excitation part of the Higgs field can be exchanged between particles as a force. In fact, I wrote a paper with Marc Sher about this once. However, the constant vacuum part of the field v does not oscillate in space or position, meaning it doesn’t carry momentum, and therefore cannot impart a force to anything.
The scalar or spin zero, positive parity Higgs acts just like the mass term in coupling an incoming left handed fermion such as a quark or electron, to an outgoing right handed fermion, and vice versa, an incoming right handed fermion to an outgoing left handed fermion. Other then this is dictated by the Dirac equation, I really don’t have a “picture” for how that works.
As I recall from the history of electromagnetic waves, people had to construct a medium in which the waves propagated, to be analogous to sound waves propagating in a gas, liquid, or solid. The think the model for electromagnetic waves had wheels and gears (I am the worst science historian.) Then that got replaced by the aether, which got thrown out by the Michelson-Morley experiment. Finally, physicists got comfortable with an empty vacuum, where changing electric fields generated changing magnetic fields in the wave, which in turn regenerated the changing electric fields.
